ELASTIC WAVE DIFFRACTION OF A PISTON
SOURCE AND ITS EFFECT ON ATTENUATION
MEASUREMENT
by
X.M. Tang, M.N. Toksoz, and C.H. Cheng
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
The radiation of an elastic wave field from a plane piston source is formulated using
the representation theorem, in which the Green's function for an elastic half space
is employed. On the basis of this formulation, we derive the radiated elastic wave
field for both compressional and shear wave sources. We study the diffraction of
elastic waves incident on a receiver that is coaxially aligned with the source. We
present a procedure in which both numerical and asymptotic techniques are employed
to allow us to evaluate the diffraction effects in any frequency range of interest. We
compare the elastic diffraction with the acoustic diffraction and find that they are
different in the near field of the piston source due to the coupling between shear and
com pressional components in the elastic case. In the far field, however, the elastic
diffraction approaches the acoustic diffraction. With the help of ultrasonic laboratory
measurements, we test the theoretical results and find that the theory and experiments
agree well. An important result of this study is that in attenuation measurements with
pulse propagation techniques where spectral ratio relative to a standard sample or ratio
of samples of the same material but of different lengths is used, it is necessary to correct
for diffraction effects. In the attenuation measurement using spectral ratio of a sample
and a standard reference sample, the attenuation can be overestimated, while in the
attenuation measurement using a spectral ratio of samples of different lengths, the
attenuation can be significantly underestimated, if corrections for diffraction effects
are not made.
INTRODUCTION
A cylindrical (or piston) shaped transducer is an important signal generating device
in many scientific applications. A common characteristic of all piston sources is the
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Tang et al.
diffraction effects in the radiated wave field. For the acoustic wave field, this phenomenon has been well studied by many authors (Williams, 1951; Bass, 1958; Gitis
and Khimunin, 1969; Harris, 1981; Tang et aI., 1988). By comparison, the nature of
the elastic wave field radiated by a planar piston has not been studied in detail. In
practice, however, many important applications, such as rock physics and nondestructive testing of solids, require the solution of the elastic problem. Needless to say, the
treatment of the elastic case is much more complicated than that of the acoustic case.
In addition to the vector nature of the elastic field, one needs to consider the vector
nature of the source vibrations, because a piston source can be made to generate either shear or compressional waves, or both. Moreover, as in the acoustic case, one also
needs to consider the arbitrarily shaped piston sources. As will be shown in the next
section, all these problems can be formulated into a compact form and solved using
the elastic representation theorem.
In the first part of this paper, we formulate the problem of elastic radiation by an
arbitrarily shaped surface stress source and calculate the radiated wave fields for both
shear and com pressional circular piston sources and their average fields incident on a
receiver that is coaxially aligned with the source. In the second part, we present a
procedure using both numerical and asymptotic methods and evaluate the diffraction
effects in the average wave field. In the third part, we give the results of the evaluations
and compare the elastic radiation with the acoustic radiation. In the final part, by
conducting a laboratory experiment, we compare our theory with experimental results
and analyze the effects of diffraction on the elastic attenuation measurements.
FORMULATION OF THE ELASTIC RADIATION OF A PISTON
SOURCE
The present problem can be formulated as the radiation of a surface distribution of
stress vibrations into an elastic half space. The frequency domain elastic wave equation
of motion for isotropic homogeneous media may be written as
(1)
where if = (Ulo U2, U3) is the displacement field, >. and p, are Lame constants, w is the
angular frequency, and p is the density of the medium. The solution of Eq. (1) for a
medium free of body forces is given by the representation theorem (Aki and Richards,
1980), which, in terms of tensor notations, can be expressed as
Uj
=
Jis
GjlUlknkdS -
Jis
UpCpklq
~~~l nk dS,
(2)
where Gjl(XbX2,X3,WjX~,X~,x~) is the tensor Green's function, Xj and xj (j = 1,2,3)
refer to the field and source points, respectively, Cpklq is the elastic tensor, Ulk is the
189
Wave Diffraction and Attenuation
given stress distribution on the surface S, and nk is the normal to this surface. In
this formulation, the summation convention is assumed and the integration is over the
source point (x~, x2' x~). If we solve the boundary value problem by finding the tensor
Green's function such that its generated traction Cpklqnk °oGt'
vanishes on S, then the
vX q
problem can be reduced to the evaluation of only the first integral in Eq. (2). For
the elastic half space where S is now the x~ = 0 plane, Johnson (1974) obtained the
Green's function for a source point located at (O,O,x~). It is easy to generalize his
Green's function for any source point (x~, x2' x~) in the half space and show that such
a Green's function is given by the following two-dimensional inverse Fourier transform:
Gjl(X},X2,X3,WjX~,X~,X~)=
\
4 1T
JOO JOO
Gjl(k1>k2,X3,W;0,0,x~)ei[kl(x,-x;)+k2(x2-x;)ldkldk2'
(3)
-00-00
where k1 , k z are the dummy wavenumber variables and GIj(k 1, k z , X3,W; 0, 0, x~) in
the integrand has been obtained by Johnson (1974, Eqs. 9 and 10). To perform the
integration in the first integral in Eq. (2), we set x~ = 0 in Eq. (3) and derive from
Johnson's Eqs. (9) and (10) the matrix expression for the two-dimensional Fourier
transform of the tensor Green's function evaluated at x~ = X2 = x~ = 0:
Ci(k1,k2,X3,W;0,0,0)
e -vX3 (-2k i '/ I
F(k)
-?k 1k z1/
fl
2,kl 1/1/'
1k 21/' ik1h)
-2k2
/.
e -V'X3
-?k21/
'kzh
- -1/':-::FC;-:(k:-:-)
2'k Z 1/1/'
1/h
fl
k~(41/1/' - h) - 1/,2h
x
where
k 1k 2(h - 41/1/')
( ik 1/'h
l
=w/a,
k = .jki + k~,
k1kz(h - 41/1/')
ki(41/1/' - h) _1/ 12 h
ik21/'h
k",
1/ = -JP k~,
F(k) = h 2 - 41/1/'k 2,
k{3 =
2ik l 1/1/12
2ik21/1/ 12
21/1/'k 2
(4)
w/fJ,
h = 2k 2 - kJ'
1/' = .jk2 - kJ'
and a and fJ are compressional and shear velocities, respectively. Once the Green's
function is known, the radiated elastic displacement can be calculated using
x
(5)
where we have used nk = Ok3 (O;j is the Kronecker delta) for the normal of S. Eq. (5),
together with the Fourier transform of the Green's function given by Eq. (4), gives
190
Tang et al.
the radiated elastic wave field for any given stress distribution CT/3 over the area 5 on
the x~ = 0 plane. In general, the surface integral (in square bracket) in Eq. (5) is
equivalent to the two-dimensional Fourier transform of the surface stress distribution.
In practical applications, there are two types of piston sources. One is the compressional source whose vibrations are normal to the surface, the other is the shear source
whose vibrations are parallel to the piston surface. We let this latter polarization be
in Xl direction. In most cases, the piston sources are circular. We now proceed to find
the radiated fields of circular compressional and shear sources. We assume a constant
stress distribution over a circular source of radius a. Thus the surface integral can be
written and integrated out as
1:1:
CT/3(w)H (a - JXi 2 + X~2) e-i(k,x; +k,x~)dxidx~ = CTdw )~J1(ka)
(6)
where H represents the step function, J1(ka) is the Bessel function, and CTI3(W) is the
source stress spectrum; I = 3 means that the stresses are normal to the x~ = 0 plane
(compressional source), and I = 1 means that the stresses are in the xi direction acting
on the x~ = 0 plane (shear source). The above integration is performed by making the
following transformations
k 1 = kcosO
r'cos</>
k 2 = ksinO
= r' sin</>
xi =
x~
(7)
Using the same transformations, but replacing xi, x~, and r' with Xl, X2, and r, we
can now simplify the integration over k 1 and k 2 in Eq. (5). We also use the following
identities
o eikrcos(O-¢) sinOcosOdO =
l~
-7rJ2(kr)sin2</>
o eikrcos(O-¢) dO
l~
27rJo(kr)
o eikrcos(O-¢) sin 2OdO =
l~
7rJo(kr)
o eikrcos(O-¢) sinOdO
l~
fo2~ eikrcos(O-¢) cos 2OdO
+ 1rJ2(kr)cos2</>
21riJ1(kr )sin</>
=
o eikrcos(O-¢)cosOdO =
l~
1rJo( kr) - 1rJ 2(kr )cos2</>
21riJ1(kr )cos</>,
(8)
where J O,J1 , and J2 are Bessel functions of the first kind and order zero, one, and two,
respectively. With the use of the transformations and identities, the two-dimensional
integration in Eq. (5) is reduced to one-dimensional integration which significantly
facilitates its numerical evaluations. Replacing X3 by z to comply with convention, we
Wave Diffraction and Attenuation
191
now express the radiated displacement fields uj(r,¢,z,w) for the compressional and
shear sources as follows:
The compressional source [I = 3 in Eq. (5)]:
UI
=
I
(k )J (k )kdk
a I r
2
roo
[2k - k~ -vz _ 2/.//./' -v'z] J
21r1-')0
F(k) e
F(k)e
I
(k )J (k )kdk
a I r
21r1-'
aoo33(w)sin¢
U2
U3
2
roo
[2k - k~ -vz _ 2/.//./' -v'z] J
)0
F(k) e
F(k)e
aoo33(w)cas¢
2
roo
[2k - k~ -vz
21r1-')0
F(k) /./e
-
aOO33(w)
=
2
2k /./
F(k)e
-v'z] JI(ka)Jo(kr)dk
. ..
(9)
The shear source [I = 1 in Eq. (5)]:
UI
=
+
U2
=
U3 =
roo [
aO"I3(W)
2k 2/./,
- F(k) e
41r1-')0
aOOI3(w)eas2¢
41r1-'
-vz + (3k 2 -
2
k~)(2k2 - k~) - 4w'k
/./'F(k)
e
roo [2k 2/./' -vz + (2k 2 -
kffi) - 4w'k2
/./'F(k)
e
F(k) e
)0
2
2
roo
[2k /./' -vz + (2k - k~) 41r1-')0
F(k)e
/./'F(k)
aOOI3(w)sin2¢
aOOI3(w)eas¢
21r1-'
4w'k2
2
roo
[_
2/.//./' -vz 2k - k~ -V'z] J
)0
F(k)e
+ F(k) e
I
e
-V'z] JI(ka)Jo(kr)dk
. ..
-V'z] J (k a )J (k r )dk
I
2
-V'z] J (k a )J (k r )dk
I
(k )J (k )kdk
a I r
2
(10)
Eqs. (9) and (10) give elastic displacement at a field point anywhere in the half space
for the compressional and shear sources. In practice, however, point measurements
are not feasible and the wave signals are often measured using a receiver of finite size.
In most cases, source and receiver are coaxially aligned, plane circular disks. It is a
common practice to define the received signal as the average wave field over the receiver
surface (Gitis and Khimunin, 1969; Tang et a!., 1988). For a receiver transducer having
the same radius as that of the source, the average field is given by:
(Uj) =
2
a
~
r" d¢ Jr uj(r,¢,z,w)rdr.
7ra J
o
(11)
o
By substitution of Eqs. (9) and (10) into Eq. (11), we obtain the averaged fields for
the two sources:
The compressional source:
(UI)
(U2)
(U3)
=
=
=
+
0
0
fooo 2k 2 -
k~ /./e _ [JI (ka)j2 dk
1r1-'
0
F(k)
ka
2
2aoo33(w)
k /./ e- 'z[JI(ka)j2 dk
1r1-'
0 F(k)
ka
_ aoo33(w)
fooo
vz
v
+
(12)
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Tang et al.
The shear source:
+
2
acr13(w)foeo k v' -vz[Jl(ka)]2dk
--e
'+
1rfL
0
F(k)
ka
2
2
acrdw) {eo (3k - 2k~)(2k2 - k~) - 4vv'k e-v'z [Jl(kaW dk
21rfL )0
v'F(k)
ka
(13)
o
=
0
It is interesting to note that the average fields are polarized in the same direction
as that of the source vibrations. A given shear or compressional source can generate
both compressional and shear signals. The first integral in either Eq. (12) or Eq. (13)
represents the compressional signal, while the second integral, the shear signal. However, as we will see in the next section, the wave amplitudes of shear waves produced
by a compressional source and the compressional waves produced by a shear source
are generally very small. Moreover, it will be instructive to compare Eqs. (12) and
(13) with their acoustic counterpart (Williams, 1951). The average particle velocity
potential (cp) is
(cp) = 2avo(w) (eo exp( -JP - k~z) [J1(kaW dk,
)0
JP - k~
ka
(14)
where k c is the acoustic wavenumber and vo(w) is the source particle velocity spectrum.
Although the integrals in Eqs. (12) and (13) appear much more complicated than the
one in Eq. (14), they have the common factors of the exponential function and the
squared Bessel function. The similarity and difference between acoustic and elastic
cases will be illustrated as we evaluate these integrals in the next section.
EVALUATION OF THE DIFFRACTION EFFECTS
To determine the diffraction effects due to the piston sources, we need to evaluate
the integrals in Eqs. (12), (13), and (14). Although approximate solutions for the
acoustic case (Eq. 14) were given by Williams (1951) and Bass (1958), it is difficult to
obtain such closed form solutions for the integrals in Eqs. (12) and (13). In addition,
approximate solutions are restricted by the assumptions under which they are valid.
For example, Bass' (1958) expression is not applicable in the near field. We will use
numerical methods to evaluate the integrals in Eqs. (12), (13), and (14). We transform
the integrals into a form that is suitable for the Gauss-Laguerre algorithm of numerical
integration.
Wave Diffraction and Attenuation
193
The integrals in Eqs. (12), (13) and (14) are in the form of
1
00
1'1=
o
f(k)
exp(
-Jp - k~z) [J,(ka)j2
k
dk,
yk
2 _ k~
a
(15)
where k-y can be each one of k"" kf3, and k c , and f(k) represents the remaining part
of the integrand in these integrals (for example, f(k) = 1 in Eq. 14). We make the
following change of variables:
Reh}
~
o.
(16)
With this change of variables, the path of integration in the complex 7 plane becomes
path G in Figure 1. It should be noted that the integrand in Eq. (15) has singularities.
For example, k = k""k = kf3,k = k c are all branch points, and the term I/F(k)
contained in f(k) (see Eqs. 12 and 13) has poles on the real k axis (Hanyga et a!.,
1985). All these singularities are mapped onto the real and imaginary axes of the
7 plane (Figure 1). Therefore, integration along this path will have to consider the
contributions from the singularities. To avoid this complexity, we make another change
of variables,
7 = ik-y + 7',
(17)
where 7' is a real variable. With this change of variables, we can deform path G into
path G', which starts from 7 = ik-y and goes to infinity parallel to the real 7 axis
(Figure 1). A careful study of the integrands of Eqs. (12), (13), and (14) indicates that
they do not have singularities between path G' and the real 7 axis. Thus the integration
along path G is equivalent to the one along path G' according to the theorem of residues
(an auxiliary path that links path G' with the real 7 axis at infinity is ignored because
of its zero contribution). Let
(18)
7 I = 7- / z,
and substituting into Eq. (15), we have
(19)
where f('y) is f(k) as the function of 7, a dimensionless dummy variable. It is worthwhile to note that the propagation factor e-ik,z appears after the above transformations. Replacing k-y by k"" kf3, and k c , we see that each integral in Eqs. (12), (13), and
(14) corresponds to the signals that propagate with compressional, shear, and acoustic velocities, respectively. The diffraction effect comes from the integral in Eq. (19),
which is to be evaluated. As clearly indicated in the equation, this integral is in the
form of Jooo g(x)e-xdx. This form of integration can be evaluated efficiently using the
Gauss-Laguerre algorithm, although g(x) is now a complex expression.
194
Tang et al.
It should be pointed out that numerical problems may arise at high frequencies for large k,(z. For large k'(z, the Bessel function Jl(~Jt2 + 2ik..,zt) approaches
Jl(~ei"/4J2k..,zt), which blows up if the complex argument of the Bessel function is
large. In practice this numerical problem often occurs at very high frequencies. This
problem can be solved if we make use of the similarity between the elastic and acoustic diffractions at large k..,z values. Bass (1958) derived an approximate solution of
Eq. (14) for the acoustic case. In terms of the integral in Eq. (15) for the acoustic case
where f(k) = ](t) = 1, this solution is given by
where E = ~(-/4a2 + Z2 - z). For large k..,z, this expression is very accurate (Gitis
and Khimunin, 1969). For the elastic cases, the second integral in Eq. (12) and the
first one in Eq. (13) approach zero as k..,z --> 00 (this will be demonstrated later), while
the first integral in Eq. (12) and the second integral in Eq. (13) can be asymptotically
evaluated by expanding ](t) in series as k..,z --> 00.
s -2 - -it- (2 s -2 - 4s -4
k",z
it
+ 8s -5) + "',
1
2 + - ( 2 - 8s- ) + ... ,
k{3z
(Eg. 13)
(Eg. 12)
(21)
where s = a/j3. Substituting Eqs. (21) into Eq. (19) and taking only the leading order,
we have the closed form asymptotic solutions for the elastic diffraction at large k..,z.
For the elastic case, the integrals represented by Eq. (15) become
lex
I{3
"" s-2Ic(z,k cx ),
kcxz --+ 00,
~ 2Ie (z, k(3),
k{3z --> 00,
(22)
where Ie is given by Eq. (20). By using the above numerical and asymptotic methods, we can evaluate the diffraction effects for different frequency range and receiver
distances.
RESULTS
We first give an example of the evaluation of the average elastic fields. In this example,
we let the compressional and shear velocities be a = 3.5 km/ sand 13 = 2.2 km/ s; we
also let lT13(W) = lT33(W) = lTo = const. (i.e., the source time function is a 8 function).
For a source transducer of radius a = 10 mm and a receiver of the same radius located
at z = 50 mm away from the source, the results of the evaluations of Eqs. (12) and
(13) in the frequency range of 0 - 1 MHz are shown in Figure 2; the solid and dashed
Wave Diffraction and Attenuation
195
curves in Figure 2a represent the com pressional and shear wave spectra generated
by a compressional piston source, while in Figure 2(b) the solid and dashed curves
represent the shear and compressional spectra by a shear piston, respectively. As can
be seen from these figures, the amplitudes of shear signals generated by a compressional
source and the compressional signals by a shear source are very small. Also, one
notices that the spectra of these signals are singular at zero frequency. Referring to
the classical Lamb's problem (Pilant, 1979), we see that these signals are called "time
integrals". This implies that in the frequency domain their spectra are divided by w,
which accounts for their singular behavior at zero frequency. Because the amplitudes
of the time integrals are small, they can be ignored in practical applications, and we
need only to consider the first term in Eq. (12) and the second term in Eq. (13).
Next we present the results of evaluation of the diffraction effects using both numerical and asymptotic methods in a broader frequency range (0 - 2.5 MHz). Using
the same parameters as those used in Figure 2, we evaluate Eq. (15) for the compressional (the first integral in Eq. 12), shear (the second integral in Eq. 13), and acoustic
(Eq. 14) cases. The results are shown in Figure 3. For comparison, the acoustic integral in Eq. (14) is multiplied by 8- 2 in Figure 3(a) and 2 in Figure 3(b), respectively
(see Eq. 22). These integrals are calculated numerically using Eq. (19) in the lower
frequency range, and asymptotically using Eqs. (20) and (22) in the higher frequency
range. In both (a) and (b) of this figure, an arrow points to the frequency point at
which the numerical and the asymptotic results are matched, and indeed the matches
are excellent. For comparison we have also plotted the evaluation of the integral in
Eq. (14) calculated entirely using Bass' formula (Eq. 20). The results are also multiplied by 8- 2 in Figure 3(a) and by 2 in Figure 3(b) for the comparison. We see that
Bass' (1958) expression is significantly different from the numerically calculated result
(dashed curves 1) near the zero frequency. This error has been pointed out by Tang
et al. (1988). One notices from both (a) and (b) of this figure, that the dashed curves
1 that are based on the acoustic theory differ from the solid curves that are based on
the elastic theory only at low frequencies. Our calculations show that this difference
decreases with increasing receiver distances. A physical explanation is that, in the
near field, the compressional or shear signal and its corresponding time integral are
intimately coupled. As these signals propagate away from the source, they become
decoupled because of their different propagation velocities and the rapid decay of the
time integral with distance and frequency (Pilant, 1979). The higher the frequency,
the faster the decoupling. This demonstrates that for large kz, the elastic diffraction
of either a com pressional or a shear wave approaches acoustic diffraction, as appears
on Figure 3. This is to be expected since in the far field, both the compressional
and shear waves obey their own scalar wave equations in an isotropic homogeneous
medium. One recalls Kirchhoff's pioneer work on the diffraction of light by a circular
aperture. Although Kirchhoff used the scalar diffraction theory to explain the behavior
of a vector field, the later vector approach of Smythe did not change his conclusion
in the axial region of the Fraunhofer zone (see Jackson, 1962). This historic example,
Tang et al.
196
together with the present example, reveals the governing principles residing in classical
continuum physics, be they electromagnetic or elastodynamic.
APPLICATION
An important application of the study of diffraction effects is in wave attenuation measurements. As we have seen in Figures 2 and 3, diffraction effects result in the decay
of wave amplitude with increasing frequency, even in the absence of intrinsic damping.
Consequently, in attenuation measurements using the amplitude decay of wave trains,
the intrinsic attenuation will be overestimated, while in the attenuation measurements
using spectral ratio methods (Toksoz et aI., 1978; Tang et aI., 1988), the diffraction effects can cause the intrinsic attenuation to be either over or underestimated, depending
on whether samples of different lengths or of different acoustic properties are used. To
illustrate this effect, let us consider the case of a wave signal generated by a cylindrical
source, propagated through a sample, and recorded by a receiver. Then the received
wave spectrum is given by
A(w) = O"(w)e-ikzG(W;a, z,a,(3)R(w),
where
(23)
u(w) is the source stress spectrum of either a compressional or shear source,
a is the transducer radius, R(w) is the instrument transfer function of the recording
system, G is the geometric spreading factor. e-ikzG is equivalent to L., in Eq. (19),
where 1.., can be either 1", or I{3 depending on whether a shear or compressional signal
is measured. The propagation factor e- ikz contains the intrinsic attenuation when the
wavenumber k is made complex,
k=w/v-i1],
where v is shear or compressional velocity and
1]
(24)
1]
is the attenuation coefficient given by
w
= 2Qv'
(25)
where Q is the quality factor of the sample. Suppose that measurements are made with
two samples using the same source and receiver. By taking the ratio of the amplitude
spectra of the two received signals and using Eqs. (23) and (24), we obtain
(26)
where subscripts 1 and 2 refer to measurements 1 and 2 and In denotes the natural logarithm. Two methods can be used to determine attenuation using Eq. (26).
The first is to use two samples of the same material with different lengths. The second is to use a reference sample (say sample 1) with known Q, a, and f3 values and
Wave Diffraction and Attenuation
197
determine the attenuation of the sample of interest relative to the reference sample.
The reference sample is chosen to be a non-attenuating material, such as aluminum,
whose Q "" 150,000 (Zemanek and Rudnick, 1961). In both cases, the attenuation of
the second sample 7)2 (or I/Q2) can be determined from the measured spectral ratio
IA2(W )1/IA1(w)1 provided that the geometric spreading term In(IG 1 1/IG21) as a function
of frequency is known (Eq. 19).
To test the elastic diffraction theory developed in this study, we carried out laboratory experiments using compressional wave sources and aluminum and lucite as
samples. Table 1 shows the properties of these materials. The samples are cylinders
with a radius of 76 mm and different lengths ranging from 24 mm to 64 mm. The
source and receiver transducers were mounted at the opposite ends. The transducers
were piezoelectric transducers, Panametrics V303 and V152, circular in shape. The
former has a radius of 7 mm, and the latter 14 mm. The center frequency is around
1 MHz. Figure 4 shows the experimental set up.
To study the effects of diffraction on the measurement of attenuation using samples
of the same material with different lengths, we measure aluminum samples of lengths
24 mm and 64 mm. Since aluminum has very low attenuation (Zemanek and Rudnick,
1961), these measurements are ideal to determine the effects of diffraction. Figure 5(a)
shows the amplitude spectra of transmitted waves for 24 mm and 64 mm long samples. The transducers are the same in both cases, with radius a = 7 mm. The different
amplitudes are due to diffraction (or spreading) effects. The In(IA 21/IA1 1) curve calculated from the spectra in Figure 5(a) is- shown in Figure 5(b) (dashed curve). This
curve has a positive slope. However, the (very small) intrinsic attenuation should
cause the In(IA 21/1A11) curve to have a slight negative slope when plotted versus frequency. Consequently, the observed effect is due to diffraction. The diffraction (or
beam spreading) effect can be corrected by the term In(IG11/IG21) in Eq. (26) using
the expression given in Eq. (19). When the correction is made, the In(lA 21/IA11) curve
is returned almost to the zero line, as shown in Figure (5)b (solid curve). Since the
aluminum has almost no intrinsic attenuation, the log amplitude ratio curve should be
almost flat, thus theory and experiment agree very well in this case.
To study the effects of transducer radius, we repeated the above experiment using
transducers of radius a = 14 mm. Figure 6(a) shows the measured wave spectra. As
compared to Figure 5(a), the larger transducers produce smaller diffraction effects.
However, the measured In(IA21/IA11) curve in Figure 6(b) still shows a non-negligible
positive slope (dashed curve). After correction, the In(IA21/IA11) curve is returned to
the zero line. There are some fluctuations about the zero line (solid curve). These
fluctuations probably imply that for large transducers, the assumed constant stress
distribution over the source surface and the constant weight average operation, as
used in our theory (Eqs. 6 and 11), may be oversimplified in the near field. (Note
that the sample is only 24 mm long, which is less than the diameter (28 mm) of the
198
Tang et aJ.
transd ucer.)
Next we study the effects of diffraction in the measurement of attenuation using
a reference sample. This technique is used extensively for laboratory measurements
(Toksoz et a!., 1978). aluminum is chosen to be the reference, while lucite is the
measurement sample. We first use the a = 7 mm transducers. Figure 7(a) shows
the measured wave spectra for the 64mm long aluminum reference and the 38mm
long lucite sample. Figure 7(b) gives the In(jA 2 1/IA1 1l curve (dashed curve) calculated
using the spectra in Figure 7(a). Then the correction term In(lG 1 1/IG21) is calculated
using Eq. (19) and used to obtain the corrected curve (solid curve). As seen from this
figure, the correction is significant, because the shear and compressional velocities of
the aluminum are much higher than those of the lucite (see Table 1). The curves in
Figure 7(b) can be linearly fitted versus frequency to give the quality factor Q of the
material (Toksoz et a!., 1978). Due to the great influence of the diffraction effects
in this particular case, the uncorrected curve gives a Q", value of 30, whereas the
corrected curve gives a Q", of 72. We repeated the lucite-aluminum experiment using
larger (a = 14 mm) transducers. Figure 8(a) shows the measured wave spectra for the
51 mm long aluminum the 38 mm long lucite dam pIes. The uncorrected and corrected
attenuation curves are shown in Figure 8(b). The linear trend of the corrected curve
becomes apparent. The linear part of this curve gives a Q", value of 76 for lucite,
agreeing with the previous estimate of Q", of 72, and the uncorrected curve gives a Q",
value of 50.
The above experiments demonstrate that the effects of diffraction are maximized
when small transducers, short samples, and samples of very different acoustic properties are used. In the attenuation measurements commonly executed in rock physics,
the diffraction effects can be minimized by using larger transducers, and by choosing a
reference sample whose acoustic properties are close to those of the measurement samples, such that the correction term In(IG1 1/IG21) can be regarded as independent of
frequency in the frequency range of measurement. In cases where the diffraction effects
are not negligible, the diffraction theory developed in this study can be used to make
the correction. Finally, when large transducers are used such that the received waves
are sensitive to the details of stress distribution over the source surface, Eq. (5) can
be used to calculate the wave radiation and diffraction provided that this distribution
can be obtained.
CONCLUSIONS
In this study, we investigated the radiation and diffraction of the elastic wave field
generated by a piston (cylindrical) source. Using the elastic representation theorem
and the corresponding tensor Green's function, we presented a formulation that can be
used to calculate the radiated elastic wave field for any given piston shape and source
Wave Diffraction and Attenuation
199
stress distribution. Assuming a constant compressional or shear stress distribution over
a circular piston surface, we derived the integral expressions for the radiated elastic
wave field. On the basis of these expressions, we obtained the average wave field
received by a receiver coaxially aligned with the source. Despite the complexity of
elastic radiation, the average field was found to polarize in the source stress direction.
For evaluating the diffraction effects, we used numerical techniques in the near field
and asymptotic techniques in the far field. It was shown that a compressional or a
shear piston generates a mainly compressional or shear signal in the axial direction,
together with some time integral signals which decay rapidly in the far field. Thus in
the far field, the radiation and diffraction of either a compressional or a shear wave are
very similar to those of an acoustic wave, despite the vector nature of the elastic waves.
In particular, we studied the effects of elastic diffraction in conjunction with elastic
attenuation measurements. The experimental results were found to agree well with
the theory, and the correction for the diffraction effects gave a reasonable estimate
of the attenuation for the material measured. Although we only did measurements
with compressional waves, the shear waves can be treated in much the same wa,y. The
knowledge obtained from this study will be useful in scientific applications where piston
sources are used.
ACKNOWLEDGEMENTS
We would like to thank Tien-When Lo for his helpful discussions and hel p in the
experimental set up. Karl Coyner of New England Research Inc. provided the samples
for the experiment. This research was supported by AFGL contract No. F19628-86K-0004, DOE grant No. DE-FG02-86ER13636, and by the Full Waveform Acoustic
Logging Consortium at M.LT.
200
Tang et at
REFERENCES
Aki, K., and Richards, P., Quantitative Seismology-Theory and Methods, W.H. Freeman and Co., San Francisco, 1980.
Bass, R., Diffraction effects in the ultrasonic field of a piston source, J. Acoust. Soc.
Am., 30, 602-605, 1958.
Gitis, M.B., and Khimunin, A.S., Diffraction effects in ultrasonic measurements, Sov.
Phys. Acoust., 14, 413-431, 1969.
Hanyga, A., Lenartowicz, E., and Pajchel, J., Seismic Propagation in the Earth, Elsevier, 1985.
Harris, G.R., Review of transient field theory for a baffled planar piston; J. Acoust.
Soc. Am., 70, 10-20, 1981.
Jackson, J.D., Classical Electrodynamics, John Wiley & Sons, Inc., New York, 1962.
Johnson, L.E., Green's function for Lamb's problem, Geophys. J. Roy. Astr. Soc., 37,
99-131, 1974.
Pilant, W.L., Elastic Waves in the Earth, Elsevier, 1979.
Tang, X.M., Toksiiz, M.N., Tarif, P., and Wilkens, R.H., A method for measuring
acoustic wave attenuation in the laboratory, J. Acoust. Soc. Am., 83, 453-462,
1988.
Toksiiz, M.N., Johnston, D.H., and Timur, A., Attenuation of seismic waves in dry and
saturated rocks: 1. Laboratory measurements, Geophysics, 44, 681-690, 1978.
Williams, A.O., Jr., The piston source at high frequencies, J. Acoust. Soc. Am., 23,
1-6, 1951.
Zemanek, J., and Rudnick, 1., Attenuation and dispersion of elastic waves in a cylindrical bar, J. Acoust. Soc. Am., 33, 1283-1288, 1961.
(
201
Wave Diffraction and Attenuation
~
Sample
I p (g/cm 3 ) I a
(m/sec)
I (3
(m/sec)
Aluminum
2.7
6410
3180
Lucite
1.2
2740
1330
~
Table I: Density p, compressional velocity a, and shear velocity (3 of the samples used
in the measurement.
202
Tang et al.
1mb}
ik..,
C'
------------------
a
b
d
c
o
C
Reb}
Figure 1: Paths of integration in the complex, plane. Path C which passes through
singularities is deformed into path C' to facilitate the numerical evaluation. The
small solid circles represent singularities. For k.., = k c , Eq. (15) has only one singularity (b) at, O. For k.., k" and k..,
k(3, Eq. (15) correspond to the integrals in
=
=
=
Eqs. (12) and (13). The singularities arq = (a) iJk~ - k&, (b) 0, (c) Jk~ - k&,
and the one furthest to the right of the real, axis, (d), which is a zero of F(k) in
Eqs. (12) and (13).
Wave Diffraction and Attenuation
203
2.00
(a)
.
~
6'1"-
~
x
_____ compresslon.l
______ !>heoar
1.50
··
~
0
~
•
1.00
W
Q
::::>
.....
...J
"'-
..,
>:
0.50
·
·
···
.
,,
,,
0.00
0.00
( b )
0.25
0.50
0.75
5.00
.
8]:t
•x •
1.00
comprploSlonal
3.75
~
0
~
2.50
w
Q
::::>
.....
...J
"'-
..,>:
1.25
0.00
0.00
···
.
--- ---- -------0.25
0.50
0.75
FREQUENCY CMH z)
Figure 2: Numerical evaluations of Eqs. (12) and (13) for a = 3.5 km/ sec, j3 =
2.2 km/ sec, z = 50 mm, and a = 10 mm. (a) Eq. (12), for the compressional
source. (b) Eq. (13), for the shear source. In (a), the solid and dashed curves are the
compressional and shear wave amplitude spectra generated by the compressional
source, while in (b), the solid and dashed curves are the shear and compressional
spectra generated by the shear source. The dashed curves correspond to the time
integrals, which decay rapidly with increasing frequency.
204
Tang et al.
Figure 3: Evaluation of Eq. (15) for the compressional, shear, and acoustic cases
using numerical and asymptotic methods. The parameters used are the same as
those in Figure 2. (a) Compressional (solid curve) versus acoustic (dashed curve
1), where the acoustic velocity equals the compressional velocity. (b) shear (solid
curve) versus acoustic (dashed curve 1), where the acoustic velocity equals the
shear velocity. In both (a) and (b), the solid curves and the dashed curves 1 are
calculated numerically in the lower frequency range, and asymptotically in the
higher frequency range. The arrows point to the frequency at which the results
of both methods are matched. The dashed curves 2 correspond to the acoustic
integrals [k c = k", in (a) and k c = k{3 in (b)] asymptotically evaluated using Bass'
expression (Eq. 20). Note that the dashed curves 1 and 2 are significantly different
at low frequencies. The acoustic curves have been multiplied by a factor of {32 / (t2
in (a) and of 2 in (b) for the comparison (see Eqs. 22).
Wave Diffraction and Attenuation
(a)
COMPARISON--ELASTIC
2.00
~
ACOUSTIC THEORIES
...• 1
_E'la'iotlc
,
_________ acoustl c
,
1.50
';::;'
0
0
X
1. DO
w
'"::::>
I-
...J
P-
0.50
~
«
o. DO
0.00
0.63
1.2S
1.88
FREQUENCY CMH z)
( b )
COMPARISON--ELASTIC
5.00
~
ACOUSTIC THEORIES
•• ,1
/".'..
12
3.75
..
_ _ elastic
,
_________ acoustl c
';::;'
0
0
X
2.50
w
'"::::>
I-
...J
P-
asymptotic
1.25
~
«
0.00
1.88
O. DO
FREQUENCY CMHz)
205
Tang et al.
206
Pulse Generator
Recording System
Sample
Source
-
I-
-
-
Receiver
Figure 4: Experimental set up. The source and the receiver are mounted at the
opposite ends of the sample. The source transducer generates a wave signal, which
propagates through the sample, and is recorded by the receiving system.
207
Wave Diffraction and Attenuation
(2~
SPECTRA OF ALUMINIUM
( a)
!.
6~
m.,,), .=7."."
1.00
Q.75
~
-"
~
•u
........
~
0.50
.....
w
'":::J>...J
".
>:
0.25
<
....
.....
0.00
0.0
( b )
......
0.5
1. a
2.0
1.5
2.00
_ _ cot"'l"'t"cteod
~
________ u nco t'" r eo c
~
E
E
~
1. 00
t pd
~
N
~
<
•
"-
E
~
0.00
-<l
~
<
:3
~1.00
-2.00
0.0
0.5
1.0
2. a
FREQUENCY (MHz)
Figure 5: (a) Amplitude spectra of compressional waves generated and received by
transducers with a = 7 mm radii. The source and the receiver are coaxially aligned,
as shown in Figure 4. The medium of propagation is aluminum. Sample lengths
are 24 mm (solid curve) and 64 mm (dashed curve). (b) The measured spectral
ratio In( IA2 1/ lAID (dashed curve) and its correction (solid curve) for the diffraction
effects. Note that the uncorrected curve has a positive slope and that the corrected
curve coincides with the zero line.
208
Tang et al.
SPECTRA OF ALUMINIUM
(a)
(2~
~
6~mm)
• • =l~mm
1.00
......... z:6'tmm
0.75
~
."
•
•
•u
0.50
W
,
,
,
I...J
"">:
(
,
,
'":::J
.
O.2~
<
0.00
0.0
( b )
1. a
0.5
1.5
2.0
2.00
_ _ co,..,..pct.d
nco r r PC: t.. d
_____ .1.1
~
E
E
1. 00
""
N
~
-
<
"-
~
E
E
0.00
----------------
'"""<
-----
~
Z
-1.00
...J
(
-2.00
o. a
0.5
1. a
I. S
2.0
FREQUENCY (MHz)
Figure 6: Same as Figure 5, but now the transducers have a = 14 mm radii.
209
Wave Diffraction and Attenuation
ALUM I NIUM (6~",,,,) ~ LUC IrE (38",,,,).
( a )
• =7",,,,
1.00
0.75
"0
•
,u•
'-'
,
,
,
,
o. so
w
,
,
,
'"::::J
I-
~
>:
0.25
'
«
..
0.00
0.0
'
.......
"
1. a
0.5
2. a
1.5
( b )
~
~
,
3.00
E
_ _ COl"rpcteod
C
,
______ . u nco I" r (I c
E
-•
t.d
1.50
~
«
"" ,
"-
0:
~
-u,
""
0.00
.. ..
~
~
~
«
~
z
·1. SO
.J
-3.00
0.0
0.5
1. a
1.5
2.0
FREQUENCY (MH z)
Figure 7: (a) Wave spectra of aluminum (z = 64 mm) (solid curve) and lucite
(z = 38 mm) (dashed curve), (b) The measured spectral ratio (dashed curve)
and its correction (solid curve) for the diffraction effects, Note that the slope of
the corrected curve is decreased,
210
Tang et al.
ALUM I N ruM
( a)
C51 mm)
!. LUC lTE C38mm). • =1 ~mm
1.00
_ _ alumInium
......... 1 UCI
te
0.15
~
."
•
,•u
0.50
w
'"
,
,
::J
f-
...J
p..
>:
0.25
.....
<
................
....
'"
0.00
0.0
0.5
1. a
1.5
( b )
e,
3.00
<
-•,•
1.50
~
<
"-
0:
~
0.00
,~
~
<
~
-1.50
_ _ COl""l"ectl?'d
Z
...J
•
-3. 00
0.0
0.5
u nc 01" I"
1.0
ec t
ed
1.5
FREQUENCY CMH z)
Figure 8: (a) Wave spectra of aluminum (z = 51 mm) (solid curve) and lucite (z =
38 mm) (dashed curve). (b) The measured spectral ratio (dashed curve) and its
correction for the diffraction effects. Note that the linear trend of the corrected
curve becomes apparent.